Evaluating the effect of environmental regulations on a closed-loop supply chain network: a variational inequality approach

Transcription

1 Evaluating the effect of environmental regulations on a closed-loop supply chain networ: a variational inequality approach E. Allevi 1, A. Gnudi 2, I.V. Konnov 3, and G. Oggioni 4 Abstract Global climate change has encouraged international and regional adoption of pollution taxes and carbon emission reduction policies. Europe has taen the leadership in environmental regulations by introducing the European Union Emissions Trading System (EU-ETS in 2005 and by developing and promoting a set of policies destined to lower carbon emissions from transport sectors. These environmental policies have significantly affected the production choices of European energy and industrial sectors. In this paper, we propose a closed-loop supply chain networ model that includes raw material suppliers, manufacturers, consumers, and recovery centers. The objective of this paper is to formulate and optimize the equilibrium state of this closed-loop supply chain networ assuming that manufacturers are subject to the EU-ETS and a carbon tax is imposed on truc transport. The model is optimized and solved by using the theory of variational inequalities. The developed model is able to capture carbon regulation, recycling, transportation and technological factors within a unified framewor and to capture their effects on production and CO 2 emission generation. Our analysis shows that the combined application of the EU-ETS at manufacturers tier and the carbon tax on truc transport implies additional costs for producers that reduce the production of goods. This has a positive outcome for environment since CO 2 emissions reduce as well. An increase of the efficiency level of the recycling process leads to an increase of the reusable raw material in the reverse supply chain. However, this production of reusable raw material can be negatively affected when a carbon tax is imposed on transport between recovery centers and manufacturers. Finally, the distance between couple of CLSC tiers plays a very important role. The lower is the distance covered by vehicles, the higher is the production of goods and the lower is the CO 2 emitted. Keywords: Closed-loop supply chain networ, environmental regulations, variational inequality approach. 1 Department of Economics and Management, University of Brescia, via S. Faustino, 74/b, Brescia 25122, Italy. elisabetta.allevi@unibs.it 2 Department of Management, Economics and Quantitative Methods, University of Bergamo, via dei Caniana, 2, Bergamo 24127, Italy. adriana.gnudi@unibg.it 3 Department of System Analysis and Information Technologies, Kazan Federal University, ul. Kremlevsaya, 18, Kazan , Russia. onn-igor@yandex.ru 4 Corresponding author. Department of Economics and Management, University of Brescia, via S. Faustino, 74/b, Brescia 25122, Italy. Phone: giorgia.oggioni@unibs.it 1

2 1 Introduction In recent years, a growing attention has been given to the impacts of climate change and carbon emissions on environment. At world-wide level, the Kyoto Protocol has been introduced in order to mitigate these effects. Other environmental policies have been applied, but only at regional level. In 2005, Europe introduced the European Union Emissions Trading System (EU-ETS that is the first and largest cap-and-trade scheme applied at international level and covers more than half of the European annual carbon emissions. In particular, the EU-ETS imposes a cap on CO 2 emissions generated by power plants and industrial installations. Each ton of CO 2 emissions has to be covered by an allowance whose price is defined on a dedicated maret. The EU-ETS has allowed Europe to maintain the Kyoto Protocol targets and now represents the main tool to achieve the ambitious target of 20% emission reductions by 2020 with respect to the 1990 level. This maret has several objectives, which are strictly related to the Energy Roadmap 2050 strategic goals identified by the European Union (EU with the aim of mitigating climate change and promoting a low carbon economy. The most ambitious EU target is an 80% decarbonization of the European energy systems by 2050 thans to investments in RES, in the transport sector, and in efficient technologies. The EU-ETS was initially subdivided in two distinct phases as indicated by Directive 2003/87/EC, but a third phase has been added by Directive 2009/29/EC. In the first two phases, that respectively covered the periods and , allowances were mostly grandfathered to the involved sectors, but this created some economic distortions as indicated in [21] and [24]. The third phase, started in 2013, substantially modifies the EU-ETS by enlarging the involved sectors and significantly reducing the amount of grandfathered allowances. This is also the tendency of the announced fourth phase that is currently under discussion. 5 Among the sector currently involved in the EU-ETS there is also aviation. According to European Commission 6, transport is the second biggest greenhouse gas emitting sector after energy and is responsible for around a quarter of European greenhouse gas emissions. Road transport significantly contributes to CO 2 emissions as well as maritime and aviation sectors. Since carbon emissions have been increasing for most of transport sectors, Europe has so far applied and intends to enhance the set of policies destined to lower emissions from road transport. These include strategies to reduce emissions from cars and vans, such as emission targets for new vehicles, and to limit fuel consumption and CO 2 emissions of heavy duty vehicles. Production processes and transport are strictly related each others and are important tiers of a supply chain networ. In addition to transport, there exist several factors that significantly affect the supply chain tiers and therefore the product flows at equilibrium. These can be classified into technological and environmental factors. The technological factors are represented by the relations between inputs and outputs of each couple of 5 See the Proposal for a Directive of the European Parliament and of the council amending Directive 2003/87/EC to enhance cost-effective emission reductions and low-carbon investments available at http: //ec.europa.eu/clima/policies/ets/revision/documentation_en.htm 6 See 2

3 tiers. The environmental factors are strictly related to the growing recognition that issues of environmental pollution accompanying industrial development should be addressed simultaneously in the operational process of supply chain management. This increasing consciousness on environmental concerns can be reflected in the application of two groups of policies that can be either applied simultaneously or in a separated way. The first group of policies is related the legislative area and it is represented by regulations that limit the carbon emission generated in the whole supply chain. The second group of policies consists in the recycling of used products that is applied to mae the production system more efficient and sustainable. In particular, some of the resources used in the production processes retain their basic physical and chemical properties during use and under the proper conditions can be recycled and reused. Businesses and organizations are increasingly modifying their supply chain models to increase sustainability in both their products and their processes. This leads to the development of closed-loop supply chain (CLSC networ where recovery centers are included in the production process. Improving the networ of supply chains remains a challenging problem both at the academic research and the practical levels. From a modeling point of view, these environmental factors can be represented by the sorting and recycling capability of recovery centers in a CLSC networ. 1.1 Paper aims A CLCS networ model is designed and managed to explicitly consider the reverse and forward supply chain activities over the entire life cycle of the product (see, e.g., [7],[8], [20], [27]. The CLSC system aims at maximizing the product value over its entire lifecycle with dynamic recovery of product parts over time. In line with European targets, product recovery and reuse improve industrial efficiency both from an economical and an environmental point of view. Moreover, the recycling process of used products (such as papers, glass, building wastes, electric, and electronic equipment in a CLSC leads to a waste reduction and to an increasing awareness of climate change problems. Parallel to the development of CLCS models that mainly focus on the recycling aspect, other academics concentrate their attention on the legislative intervention applied in a supply chain networ to reduce carbon emissions. This has contributed to the develop of green-supply chain management (GrSCM models (see [4], [9], [19], [23], [25], [28], [29]. In this paper, we integrate all these environmental issues in one single model and we investigate how the joint application of carbon regulation (legislative intervention, recycling, technological and transport factors impact on the product flows of a CLSC networ. More precisely, this paper aims at investigating the following issues, taing into account the technological and the material balance constraints that characterize each tier of the supply chain networ: Q1: How the application of carbon reduction policies both at production and transportation levels can affect the product flows and carbon emission generation in a CLSC; Q2: How the application of environmental policies can affect the recycling process in the CLSC; 3

4 Q3: Whether the transport distance between couples of CLSC tiers can affect product flows when carbon regulations are applied. To this scope, we develop a CLSC model where manufacturing processes are subject to a restrictive EU-ETS, namely allowances are assumed to be fully auctioned. 7 Moreover, in order to account for the European Commission s willingness to regulate carbon emissions from heavy-duty vehicles, we impose a carbon tax on CO 2 generated by trucs. Unle the papers quoted above, our model is based upon variational inequality theory, which facilitates the formulation of equilibrium problems that describe the interactions of several agents whose choices are subject to technological, economical and environmental constraints. The spatial control of such systems requires to consider not only the reactions of separate marets (or agents, but also technical constraints such as production capacity and mass balance constraints. Variational inequalities were developed and adopted to study various inds of equilibria such as spatial equilibrium models ([14], financial networs (see, e.g., [1], [15] and transportation networs (see, e.g., [2], [22]. In addition to these applications, there exist papers that study supply chain management such as [16] and [31]. The variational inequality theory was also applied to analyze the effects of the application of a carbon tax in a power supply chain networ (see [30] or to model CLCS networ for the electrical and electronic equipment waste Directive applied in Europe (WEEE Directive 2002/96/EC as in [10], [17] and [26] or to develop sustainable supply chain models as in [18]. However, to the best of our nowledge, it is the first time that that variational inequality approach is applied to develop a model that taes into carbon regulation, recycling, transportation and technological factors within a unique framewor. Moreover, the existing papers in the field of variational inequalities do not investigate the impacts of the combined application of the EU-ETS on manufacturing and of a carbon tax on transport in a CLCS networ as we do (see Figure 1 for the considered environmental factors. We also notice that variational inequality models admit many efficient solution methods, which can be easily implemented (see [5] and [13]. 1.2 Contributions Our model taes into account several issues (carbon regulation, recycling, endogenous demand in a unified variational inequality model framewor, while the recent literature usually threats them separately. The main contributions of this wor are the following: 1. To analyze the application of carbon regulation both on manufacturing (EU-ETS and transport and to evaluate the effect of their combined application on product flows; 2. To propose a possible policy, still under discussion at European level, that can be applied to limit CO 2 emissions of heavy-duty vehicles; 7 Note that this assumption can be easily adjusted when modeling sectors that still receive free allowances in the third EU-ETS phase. For more details, see htm 4

5 Environmental factors Recycling Ecological good Non-ecological good Carbon Legislative intervention EU-ETS Carbon tax on transport Figure 1: Considered environmental factors 3. To investigate the impacts of the joint application of legislative intervention and recycling in a complex CLSC networ; 4. To develop an adequate variational inequality model of a CLSC subject to carbon reduction regulations. The paper is organized as follows. Section 2 describes the proposed CLSC networ under environmental regulations whose model is presented in Section 3. Section 4 derives the equilibrium conditions of the CLSC networ model and illustrates the qualitative properties of the corresponding variational inequality model. Section 5 describes the algorithm used to implement the variational inequality model, while Section 6 is devoted to the descriptions of the results of the numerical experiments. Finally, Section 7 concludes the paper with some final remars. 2 A CLSC networ under environmental regulations This paper aims at analyzing different environmental issues in one single CLSC networ model. In particular, we combine the recycling aspects that are typical of a CLSC with environmental policies applied at different tiers of the supply chain. As indicated above, we assume that manufacturers are subject to the EU-ETS system and a carbon tax is imposed on CO 2 emissions generated by heavy-duty vehicles (trucs. We propose a CLCS networ where raw material suppliers, manufacturers and demand marets participate to the forward logistics; while, in the reverse logistics, recovery centers collect used but recyclable products from demand marets. These used products are thus dissembled and transformed in reusable raw materials that are sold bac to manufacturers. Figure 2 illustrates this CLSC networ. We assume that, in this CLSC networ, manufacturers produce two goods with the same end-use that only differ for the production processes and the raw material used. More precisely, one good is produced using reusable raw materials that manufacturers buy from recovery centers after the recycling process, while the other is produced using virgin (new raw materials that manufacturers purchase from raw material suppliers. In 5

6 FORWARD SUPPLY CHAIN REVERSE SUPPLY CHAIN 1 n N Raw material suppliers 1 i I Manufacturers 1 m M Recovery centers 1 i I Manufacturers 1 K Demand marets 1 K Demand marets Landfill Figure 2: CLSC networ the following, we label the first good as ecological" or, in a simpler way, as eco, while the second is denoted as non-ecological or, for the sae of simplicity, as neco. We assume that, in the forward logistics, manufacturers produce two goods with the same end-use that differ for the production processes and the raw material used. More precisely, one good is produced using reusable raw materials that manufacturers buy from recovery centers after the recycling process, while the other is produced using virgin (new raw materials that manufacturers purchase from raw material suppliers. We label the first good as ecological or, in a simpler way, as eco, while the second is denoted as non-ecological or, for the sae of simplicity, as neco. Since the production processes applied to produce ecological and non-ecological goods are different, we assume that the production costs and thus the prices applied to the two goods are not the same. However, besides these production differences, the two goods are similar and can be consumed in the same way. An example of this can be represented by the recycled paper and the not-recycled paper products which have the same end-use but are characterized by different production processes and costs. Manufacturers sell their goods to consumers located in different demand marets. Note that both manufacturers and consumers are involved in the reverse logistics in the following way. Recovery centers collect used but recyclable products at demand marets and transform them in reusable raw materials that are then purchased by manufacturers. In the transformation process operated in the recovery center, it may happen that part of the used products could not be recyclable. These useless disassembled materials are sent directly to the landfill. The raw material suppliers and the recovery centers respectively sell virgin and reusable raw materials to manufacturers. It is also assumed that agents in one tier of this CLSC act independently of agents operating in another tier of the same supply chain. Moreover, inside their own tier, agents are assumed to operate in a non-cooperative fashion. 6

7 FORWARD SUPPLY CHAIN REVERSE SUPPLY CHAIN Raw material supplier Manufacturers CO 2 CO 2 Recovery centers Manufacturers CO 2 Landfill Demand marets CO 2 Demand marets Figure 3: Environmental regulations applied in the CLSC networ As indicated above and here depicted in Figure 3, we assume that production activity at manufacturers level is subject to the EU-ETS, implying a carbon charge to manufacturers that have to cover each ton of CO 2 emitted through an allowance. In addition, in order to model the European Commission s willingness to regulate carbon emissions from heavyduty vehicles, we impose a carbon tax on CO 2 generated by trucs during the transport phase. Following [19], we suppose that trucs are only used in the following phases: transferring of virgin raw materials from suppliers to manufacturers (forward logistics; transferring of ecological and non-ecological goods from manufacturers to consumers at demand marets (forward logistics; transferring of reusable raw materials from recovery centers to manufacturers (reverse logistics. As in Pasoy et al. (2011, we assume that transfers of used product from demand marets to recovery centers and of useless materials from recovery centers to landfill are conducted with vans of smaller dimensions that are not environmentally regulated. 3 Modeling the CLSC networ under environmental regulations In this section, we develop the CLSC networ model with raw material suppliers, manufacturers, demand marets, and recovery centers as indicated in Figure 1. We first 7

8 focus on the forward logistics and we optimize the behaviour of raw material suppliers, manufacturers, and consumers at demand marets. We then consider the reserve logistics and we describe the optimization problem solved by recovery centers and the role played by demand marets and manufacturers in the recycling process. The optimality conditions of each of the agents problem are derived using the variational inequality theory. Finally, Section 4 defines the equilibrium conditions of the whole CLSC networ model. 3.1 Notation Indices N: number of raw material suppliers in the CLSC networ, n = {1,..., N}; V : number of virgin raw materials in the CLSC networ, v = {1,..., V }; R: number of reusable raw materials in the CLSC networ, r = {1,..., R}; I: number of manufacturers in the CLSC networ, i = {1,..., I}; K: number of demand marets in the CLSC networ, = {1,..., K}; M: number of recovery centers in the CLSC networ, m = {1,..., M}; T : number of trucs in the CLSC networ, t = {1,..., T }. Parameters: unit transportation costs in the forward chain ρ vnit : unit truc transportation cost of virgin raw material v during the transportation from supplier n to manufacturer i using truc t; ρ rmit : unit truc transportation cost of reusable raw material r during the transportation from recovery center m to manufacturer i using truc t; ρ t : unit truc transportation costs of truc t during the transportation from manufacturer i to demand maret. Parameters: unit transportation costs in the reverse chain (faced by recovery centers ρ m : unit transportation cost of used product from demand maret to recovery center m; ρ m : unit transportation cost of useless disassembled materials from recovery center m to landfill. Parameters: transportation capacity in the forward chain z nit : transportation capacity of truc t from supplier n to manufacturer i; z mit : transportation capacity of truc t from recovery center m to manufacturer i; z t : transportation capacity of truc t from manufacturer i to demand maret ; 8

9 Parameters: percentage rates in the reverse chain β rm : percentage of reusable raw material r that recovery center m is able to extract from used product collected in the demand marets. Note that (1 R r=1 β rm represents the proportion of useless materials got in the transformation process of used product that recovery center m is not able to recycle and thus it sends to landfill; β m : percentage of useless materials got in the transformation process of used product that recovery center m is not able to recycle and thus it sends to landfill. Note that β m = 1 R r=1 β rm; θ eco : return ratio of the ecological good used in demand maret ; θ neco : return ratio of non-ecological good used in demand maret ; Parameters: subsidies s m : unit of subsidy received by recovery center m for recycling used product collected at demand marets; Parameters: landfill costs ω m : cost per unit of useless disassembled material that recovery center m send to landfill. Parameters: CO 2 regulations α: carbon tax on trucs; τ t : carbon emission factor of truc t; p CO 2 : carbon price; φ eco i : carbon emission factor of manufacturer i for producing the ecological good; φ neco i : carbon emission factor of manufacturer i for producing the non-ecological good; Parameters: output production capacity constraints Q eco i : manufacturer i s production capacity of ecological good; Q neco i : manufacturer i s production capacity of non-ecological good; Additional parameters x ni : distance (in Km from supplier n to manufacturer i; x : distance (in Km from manufacturer i to demand maret ; x mi : distance (in Km from recovery center m to manufacturer i; x m : distance (in Km from demand maret to recovery center m; x m : distance (in Km from recovery center m to landfill; 9

10 Variables: forward chain q rmit : nonnegative amount of reusable raw material r that recovery center m provides to manufacturer i with truc t to produce the ecological good. We group the shipment of all the reusable raw materials into the column vector Q R R RMIT ; q vnit : nonnegative amount of virgin raw material v that supplier n provides to manufacturer i with truc t to produce the non-ecological good. We group the shipment of all the virgin raw materials into the column vector Q V R V NIT ; qt eco : nonnegative amount of ecological good that manufacturer i sells and transfers to demand maret with truc t. We group the shipment of all the ecological goods from all manufacturers to all demand marets into the column vector Q eco R IKT ; qt neco : nonnegative amount of non-ecological good that manufacturer i sells and transfers to demand maret with truc t. We group the shipment of all the non-ecological goods from all manufacturers to all demand marets into the column vector Q neco R IKT ; Variables: reserve chain q m : nonnegative amount of recycling output collected in demand maret and sent to recovery center m. We group the volume of recycling products between all demand marets and recovery centers into the column vector Q R KM. Note that in the following variables with " indicate equilibrium solutions. 3.2 Raw material suppliers behaviour and their optimality conditions In the forward CLSC, suppliers provide virgin raw materials to manufacturers using trucs. Let c vn denote the procurement cost of virgin raw material v faced by supplier n, which is a continuous and convex function and depends on the virgin raw material quantity. In particular, one has: ( c vn = c vn q vnit v, n (1 i=1 The total profits that each raw material supplier aims at maximizing are given by: Z n = V v=1 i=1 p vni q vnit V v=1 c vn ( i=1 α i=1 q vnit τ t x ni V v=1 V v=1 i=1 q vnit x ni ρ vnit q vnit (2 10

11 subject to z nit V q vnit 0 n, i, t (γ nit (3 v=1 q vnit 0 v, n, i, t (4 Profits (2 result from the difference between the revenues ( V I v=1 i=1 p vni q vnit accruing from selling virgin raw materials q vnit to manufacturers at optimal price p vni and the costs represented by the raw material procurement burdens ( V v=1 c vn( I T i=1 q vnit, the truc transportation charges ( V I T v=1 i=1 x ni ρ vnit q vnit and the carbon tax related to transport (α I T i=1 τ t x ni V v=1 q vnit. This tax α is imposed on each ton of CO 2 emitted by trucs when transferring virgin raw material from suppliers to manufacturers. The total amount of emission generated depends on the emission factor of each truc τ t and is proportional to the quantity of raw materials transported ( V v=1 q vnit and the covered distance (x ni. Note that virgin raw material transportation is limited by constraint (3 and the non-negativity constraint (4 is imposed on variable q vnit. Let assume that virgin raw material suppliers compete in a non-cooperative fashion. The optimality conditions for all suppliers n can be thus described simultaneously using the following variational inequality: determine the solution (Q V, γ V NIT NIT R satisfying: N V n=1 v=1 i=1 N n=1 i=1 ( I c T vn p i=1 q vnit vni ρ vnit x ni α x ni τ t γnit [q vnit qvnit] (5 [ V z nit v=1 q vnit ] q vnit [γ nit γ nit] 0 (Q V V NIT NIT, γ R Note that γ nit is the Lagrangian multiplier associated with constraint (3, while γ R NIT. 3.3 Manufacturers behaviour and their optimality conditions In the forward CLSC, manufacturers produce both the ecological and the non-ecological goods that have the same end-use, but differ for the production processes and the raw material used. The ecological good is produced using reusable raw materials that are provided by recovery centers after the recycling process, while the the non-ecological is produced using virgin raw materials that manufacturers buy from raw material suppliers. Since the production processes applied to produce ecological and non-ecological goods are different, we assume that the production costs and thus the prices applied to the two goods are not the same. However, besides these production differences, the two goods are similar and can be consumed in the same way. We therefore introduce the following function defining the manufacturing costs. Let c eco denote the manufacturing cost faced by manufacturer i for producing the ecological good qt eco using reusable raw material that is sold to demand maret. This cost is a 11

12 continuous and convex function and depends on qt eco. In particular, one has: ( T c eco = ceco q eco t i, (6 Let c neco denote the manufacturing cost faced by manufacturer i for producing the nonecological good qt neco using virgin raw materials that is sold to demand maret. This cost is a continuous and convex function and depends on qt neco. In particular, one has: ( T c neco = c neco qt neco i, (7 We also assume that manufacturers face additional costs to further refine the reusable raw materials that they receive from recovery center. These costs are defined by the continuous and convex function c ri that depends on the quantity of reusable raw material q rmit that manufacturer i buy from recovery center m (see Section 3.5. In particular, one has: ( M c ri = c ri q rmit i, r (8 m=1 The total profits that each manufacturer i aims at maximizing are given by: Z i = R =1 r=1 m=1 p CO2 subject to p eco M p rmi ( R M c ri r=1 (φ eco i =1 q eco t K =1 q rmit m=1 =1 c eco q rmit p neco ( T =1 q neco t q eco t qt eco φneco i qt neco α Q eco i Q neco i =1 =1 =1 q eco t R q eco t V v=1 n=1 =1 N p vni c neco ( T q vnit (9 q neco t x ρ t (q eco t qneco t =1 x τ t (q eco t qneco t 0 i (µeco i (10 q neco t 0 i (µ neco i (11 M r=1 m=1 12 q rmit i (η eco i (12

13 =1 q neco t V N v=1 n=1 q vnit i (η neco i (13 z t (q eco t qneco t 0 i,, t (δ t (14 q vnit 0 v, n, i, t (15 q rmit 0 r, m, i, t (16 q eco t 0 i,, t (17 q neco t 0 i,, t (18 Objective function (9 states that manufacturers profits are given by the difference between revenues accruing from selling the ecological ( K T =1 peco qt eco and the non-ecological ( K T =1 pneco qt neco goods at prices p eco and p eco respectively to consumers in demand marets and all the operating, transportation and carbon costs. The operating costs are represented by the manufacturing costs ( K =1 ceco ( T qeco t and K =1 cneco ( T qneco t and the charges to refine the reusable raw materials ( R r=1 c ri( M T m=1 q rmit. The term T K =1 x ρ t (qt eco qneco t indicates the total transportation costs. We assume that manufacturers face carbon costs both at production and transport levels. More precisely, manufacturers production activity is subject to the EU-ETS. For this reason, each ton of CO 2 emission generated in the production process has to be covered by an allowance. The carbon costs due to the EU-ETS are expressed by the term p CO2 T K =1 (φeco i qt eco φneco i qt neco where p CO 2 indicates the exogenous allowance price and T K =1 (φeco i qt eco φneco i qt neco stands for the total carbon emissions generated in producing ecological and non-ecological goods. As for suppliers that use trucs to transfer virgin raw materials, manufacturers are subject to a carbon tax α when transferring ecological and non-ecological goods to consumers located in the demand marets. The total amount of emission generated depends on the emission factor of each truc τ t and is proportional to the quantity of transported goods (qt eco qneco t and the covered distance (x. Note that the manufacturers have to consider a series of constraints while maximizing their profits. In addition to the variable non-negativity constraints (15-(18, they have to account for the production capacity constraints (10-(11; the production balance constraints (13-(12 and the truc transportation constraint (14. Let assume that manufacturers behave in a non-cooperative fashion. Therefore, the optimality conditions for all manufacturers i can be described simultaneously using the following variational inequality: determine the solution (Q V, Q R, Q eco, Q neco, µ eco, µ neco, η eco, η neco, δ R satisfying R M r=1 m=1 i=1 V N v=1 n=1 i=1 V NIT RMIT IKT IKT IIIIIKT ] [p vni ηi eco [q vnit qvnit] (19 ( M c T ri p m=1 q rmit rmi η neco [q rmit qrmit] q rmit i 13

14 i=1 i= i=1 i= p eco p neco i=1 i=1 i=1 ( c eco T qeco t q eco t ( c neco T qneco t [ Q eco i [ Q neco i r=1 n=1 q neco t =1 =1 q eco t q neco t [ R N qrmit i=1 v=1 n=1 p CO2 φ eco i x ρ t α x τ t µ eco i η eco i δ t ] [q eco t q eco t ] p CO2 φ neco i x ρ t α x τ t µ neco i ] ] =1 [ V N qvnit i=1 =1 [z t (q eco t =1 [µ eco i [µ neco i q eco t q neco t ] ] η neco i δ t µ eco i ] µ neco i ] [η eco i [η neco i ] η eco i ] η neco i ] ] qt neco [δ t δt] 0 [q neco t q neco t ] (Q V, Q R, Q eco, Q neco, µ eco, µ neco, η eco, η neco V NIT RMIT IKT IKT IIIIIKT, δ R Note that µ eco i, µ neco i, ηi eco, ηi neco, and δ t are the Lagrangian multipliers of constraints (10, (11, (13, (12, and (14 respectively, while µ eco, µ neco, η eco, η neco are column vectors of all manufacturers multipliers and δ R IKT. 3.4 Consumers behaviour and demand maret equilibrium conditions Consumers participate to both the forward and the reverse CLCS. In the forward CLCS, consumers decide the quantity of ecological and non-ecological goods that they desire to buy from each manufacturer i and how much they are willing to pay for them. In the reverse CLSC, consumers provide used products that has to be sent to recovery centers. In the forward CLSC, the equilibrium conditions for consumers at the demand maret are as defined in Nagurney et al. (2002. In particular, let d eco and p eco respectively denote the quantity of ecological good, produced with reusable raw material, demanded by consumers at demand maret and the price that they are willing to pay. Let assume that demand d eco is a continuous function of price p eco and that p eco R K. We define: d eco = d eco (peco (20 In a similar way, let d neco and p neco respectively denote the quantity of non-ecological good, produced using virgin raw material, demanded by consumers at demand maret and the 14

15 respective price that they are willing to pay. Let assume that demand d neco is a continuous function of price p neco and that p neco R K. We define: d neco = d neco (p neco (21 In addition, we assume that there are transaction costs related to the purchase of the ecological and non-ecological goods. Let tc eco and tcneco denote the transaction costs between manufacturer i and consumers at demand maret associated with the purchase of ecological qt eco and non-ecological qneco t goods respectively. These transaction costs are assumed to be continuous and positive and are defined as follows: for all manufacturers i = 1,..., I tc eco tc neco = tceco = tc neco ( T ( T q eco t q neco t i, (22 i, (23 On this basis, (24 and (25 represent the equilibrium conditions for consumers at the demand maret that buy the ecological goods. In particular, for all manufacturers i = 1,..., I, it holds that: p eco tc eco ( T q eco t = p eco p eco if if T qeco t > 0 T qeco t = 0 (24 d eco (peco = I T i=1 qeco t if p eco > 0 I T i=1 qeco t if p eco = 0 where p eco is the price applied by manufacturers to the ecological goods (see Section 3.3. Condition (24 indicates that consumers at demand maret buy the ecological good from manufacturer i if the charged price p eco increased by the transaction costs tc eco does not exceed the price p eco that they are willing to pay for the ecological good. On the other side, condition (25 states that, if price p eco is positive, then the ecological consumed at demand maret corresponds exactly to the consumers demand in that maret. In a similar way, (26 and (27 correspond to the equilibrium conditions for consumers at the demand maret that purchase the non-ecological goods. In particular, for all manufacturers i = 1,..., I, it holds that: (25 p neco tc neco ( T q neco t = p neco p neco if if T qneco t > 0 T qneco t = 0 (26 15

16 (p neco d neco = I T i=1 qneco t if p neco > 0 I T i=1 qneco t if p neco = 0 where p neco is the price applied by manufacturers to the non-ecological goods (see Section 3.3. Consumers also participate to the reverse CLCS networ. After the consumption process, both the ecological and non-ecological goods, or part of them, can be recycled and reused. We assume that these used products are collected by consumers and sent to recovery centers in order to be transformed in reusable raw materials. For each unit of used product collected in demand maret and sent to recovery centers (independently of the raw materials used to produce them, consumers receive a buy-bac price equal to p m from recovery centers. Note that transportation costs are faced by recovery centers. Consumers aversion is modeled by a continuous function a (Q that depends on the amount of used product returned to all recovery centers. The larger is the quantity of collected products, the more a recovery center has to offer as a buy-bac price. The increase of the buy-bac price represents an incentive for consumers to recycle. This means that the buy-bac price p m splits consumers into two groups: those who are willing to recycle and return used products to recovery centers and those are not. In addition, the amount that recovery centers have to pay-bac to consumers at demand marets not only depend on the quantity of used product they desire to collect, but also on the quantity collected by their competitors (see [10]. To sum up consumers behaviour in the reverse CLSC can be defined by condition (28 subject to constraint (29. = p a (Q m if q m > 0 (28 p m if q m = 0 subject to (27 M q m m=1 i=1 θ eco q eco t i=1 θ neco q neco t (σ (29 Condition (28 state that consumers at demand maret will choose to return a volume of product corresponding to the value of the buy-bac price. Constraint (29 imposes that the quantity of outputs returned to recovery centers by consumers at demand maret is not greater than the sum of the ecological and non-ecological goods purchased by manufactures in the forward chain multiplied by the respective return ratios. Considering consumers behaviour both in the forward and the reverse CLSC, their optimality conditions at demand marets are defined as follows: determine (Q eco, Q neco, Q, p eco, p neco, σ IKT IKT KMKKK R satisfying 16

17 [ ( T p eco tc eco i=1 =1 [ i=1 =1 =1 m=1 =1 p neco tc neco q eco t ( T q neco t p eco θ eco p neco M [a (Q p m σ] [q m qm] [ =1 i=1 [ =1 i=1 [ i=1 q eco t q neco t θ eco d eco (p eco ] d neco (p neco q eco t i=1 [p eco ] θ neco p eco ] [p neco q neco t σ ] θ neco p neco ] M m=1 q m [q eco t q eco t ] (30 ] σ ] [q neco t [σ σ ] 0 (Q eco, Q neco, Q, p eco, p neco IKT IKT KMKKK, σ R q neco t ] Note that σ is the Lagrangian multiplier associated with constraint (29, while σ R K. 3.5 Recovery centers behaviour and their optimality conditions Recovery centers operate in the reverse CLSC. As explained in Section 3.4, recovery centers pay a buy-bac price p m to consumers for the used and recyclable products that are collected at demand marets. This represents a cost for recovery centers that is proportional to the collected amount of used products. Once received these used products from the demand marets, recovery centers have to transform them in order to get reusable raw materials that can be sold to manufacturers. Let c rm denote the recycling costs faced by recovery center m to transform the used product collected at demand marets into reusable raw material r. These costs are continuous and convex and are proportional to the amount of used products that the recovery center receives from consumers and are given by: c rm = c rm (β rm q m r, m (31 =1 The recycling activity is subsidized and, for each unit of used product collected at demand marets, recovery centers receive a subsidy equal to s m. Not all of the collected used products can be transformed into reusable raw materials, i.e. part of the disassembled materials obtained in the transformation process cannot be used again and thus is sent to a landfill. The amount of useless materials that are sent to landfill corresponds to the difference between the used products collected at demand marets and the total amount of reusable raw materials that recovery centers obtain in the recycling process: R β m q m = (1 β rm =1 17 r=1 =1 q m

18 As indicated in constraint (33, the transformation process of used product allows recovery centers to get reusable raw materials q rmit that are then sold to manufacturers at price p rmi (compare Section 3.3. Each recovery center m aims at maximizing its profits as defined below: Z m = subject to i=1 R r=1 i=1 r=1 p rmi q rmit s m q m p m q m (32 =1 =1 =1 R c rm (β rm q m ω m β m q m x mi ρ rmit =1 R q rmit α r=1 i=1 =1 x mi τ t x m ρ m q m x m ρ m β m i=1 =1 q m R q rmit q rmit β rm q m r, m (ϕ rm (33 z mit =1 r=1 R q rmit 0 m, i, t (λ mit (34 r=1 q rmit 0 r, m, i, t (35 q m 0, m (36 Profits (32 are given by the difference between the revenues ( R I T r=1 i=1 p rmi q rmit arising from selling to manufacturers reusable raw materials ( T q rmit at price p rmi plus subsidies (s m K =1 q m and costs. These costs are associated to the transformation and recycling process ( R r=1 c rm(β rm K =1 q m, the buy-bac price ( K =1 p m q m, the landfill charges (ω m β m K =1 q m and all transportation fees, including CO 2 taxes (α T I i=1 x mi τ t R r=1 q rmit. In maximizing their profits, recovery centers have to account for some operating constraints. Constraint (33 defines the amount of reusable raw materials that recovery center obtain after the transformation process of used products. Constraint (34 imposes a transportation limit on trucs used to transfer reusable raw materials from recovery center m to manufacturer i. Finally, non-negativity constraints (35-(36 are imposed on problem variables. Let assume that recovery centers behave in a non-cooperative fashion. Assuming that the recycling costs faced by recovery center are a continuous and convex function, the optimality conditions of all recovery centers can be described simultaneously using the following variational inequality: determine (Q R, Q, ϕ, λ RMIT KMRMMIT R satisfying 18

19 [ M R s m p m =1 m=1 R M r=1 m=1 i=1 r=1 m=1 r=1 c rm(β rm K =1 q m q m ω m β m x m ρ m (37 x m ρ m β m β rm ϕ rm] [qm q m] [ p rmi x mi ρ rmit α x mi τ t ϕ rm λ mit] [q rmit qrmit] [ R M β rm qm M m=1 i=1 =1 [ R z mit r=1 i=1 q rmit ] q rmit ] [ϕ rm ϕ rm] [λ mit λ mit] 0 (Q R RMIT KMRMMIT, Q, ϕ, λ R Note that ϕ rm and λ mit are the Lagrangian multipliers associated with constraint (33 and (34 respectively, while ϕ R RM and λ R MIT. 4 Equilibrium conditions and qualitative properties of the CLSC networ model In the forward logistics, in equilibrium, the shipments of virgin and reusable raw materials respectively from suppliers and recovery centers to manufacturers must be equal to the shipments that manufacturers accept from virgin raw material suppliers and recovery center respectively. In a similar way, the amounts of ecological and non-ecological goods purchased by consumers at demand marets must be equal to the amounts sold by manufacturers. In the reverse logistics, in equilibrium, the amount of used products that consumers at demand marets sell to recovery centers must be equal to the amount bought by recovery centers. Analogously, the amount of useless materials that recovery centers send to landfill must be equal to the amount that landfill accept. The equilibrium shipment and price pattern in the CLSC must satisfy the sum of variational inequalities (5, (19, (30, and (37 in order to formalize the agreements between the tiers. This is formally stated in Definition 1. Definition 1 (Equilibrium. The equilibrium state for the closed-loop supply chain networ is the one where flows between reverse and forward tiers coincide and satisfy the sum of variational inequalities (5, (19, (30, and (37. The variational inequality that governs equilibrium conditions according to Definition 1 is now derived. Theorem 1 (Variational Inequality Formulation of the closed-loop supply chain networ. The equilibrium condition non capital governing the closed-loop supply chain networ with material transformation factors, recycled and new (virgin materials according to Definition 1 coincides with the solution to the following variational inequality: 19

20 Find (Q R, Q V, Q eco, Q neco, Q, p eco, p neco, γ, µ eco, µ neco, η eco, η neco, δ, σ, ϕ, λ RMIT V NIT IKT IKT KMKKNIT IIIIIKT KRMMIT R such that: ( Mm=1 R M c T ri q rmit x mi ρ rmit α x mi τ t ϕ rm r=1 m=1 i=1 q λ mit ηeco i [q rmit q rmit ] (38 rmit ( Ii=1 T N V cvn q vnit x ni ρ vnit α x ni τ t γ nit n=1 v=1 i=1 q ηneco i [q vnit q vnit ] vnit ceco tc eco i=1 i= i=1 i= cneco ( T qt eco q t eco ( T qt neco q t neco M a (Q σ sm R =1 m=1 r=1 tc neco ( T qt eco q t eco ( T qt neco q t neco p CO2 φ eco i x ρ t α x τ t µ eco i η eco i δ t peco θ eco σ ] [q eco t qeco t ] p CO2 φ neco i x ρ t α x τ t µ neco i η neco i δ t pneco c rm(β rm K =1 q m q m ω m β m x m ρ m θ neco σ ] [q neco t q neco t ] x m ρ m β ] m β rm ϕ rm [qm q m ] q eco t d eco (peco [p eco p eco ] q neco t d neco (p neco [p neco p neco ] =1 i=1 =1 i=1 N V z nit q vnit [γ nit γ nit ] n=1 i=1 v=1 Qeco i q eco t [µ eco i µ eco i ] Qneco i q neco t [µ neco i µ neco i ] i=1 =1 i=1 =1 R N q rmit K V N q eco t [η eco i η eco i ] q vnit K q neco t [η neco i η neco i ] i=1 r=1 n=1 =1 i=1 v=1 n=1 =1 [z t (q eco t q neco ] t [δ t δ t ] K M θ eco q eco t θ neco q neco t q m [σ σ ] i=1 =1 =1 i=1 i=1 m=1 R M β rm q I m q rmit [ϕ rm ϕ rm ] M R z mit q rmit [λ mit λ mit ] 0 r=1 m=1 =1 i=1 m=1 i=1 r=1 (Q R, Q V, Q eco, Q neco, Q, p eco, p neco, γ, µ eco, µ neco, η eco, η neco, δ, σ, ϕ, λ K where Ω {(Q R, Q V, Q eco, Q neco, Q, p eco, p neco, γ, µ eco, µ neco, η eco, η neco, δ, σ, ϕ, λ (Q R, Q V, Q eco, Q neco, Q, p eco, p neco, γ, µ eco, µ neco, η eco, η neco RMIT V NIT IKT IKT KMKKNIT IIIIIKT KRMMIT, δ, σ, ϕ, λ R } Proof. See [16]. Note that variational inequality (38 can be rewritten in the standard form (see [14] as follows: find X Ω, such that: where F (X, X X 0, X Ω (39 X (Q R, Q V, Q eco, Q neco, Q, p eco, p neco, γ, µ eco, µ neco, η eco, η neco, δ, σ, ϕ, λ with 20

21 V MIT RNIT IKT IKT KMKKNIT IIIIIKT KRMMIT X R and F (X (F rmit, F vnit, Ft eco, F t neco, F m, F eco, F neco, F nit, F µ,eco i, F µ,neco i, F η,eco i, F η,neco F t, F, F rm, F imt with the specific component of F (X being given by the respective functional terms preceding the multiplication signs in (38. i, For the sae of simplicity, in the following we will indicate the standard variational inequality (39 when referring to variational inequality (38 defining the equilibrium of our model. Following [16], it is possible to retrieve prices p vni, p rmi, peco, p neco, and p m for v = 1,..., V ; r = 1,..., R; n = 1,..., N; i = 1,..., I; = 1,..., K; m = 1,..., M from the solutions of variational inequality (38 (or (5, (19, (30, and (37. For instance, consider the price p vni (v = 1,..., V ; n = 1,..., N; i = 1,..., I in problem (2-(4. Since the objective function (2 is continuously differentiable and concave and the feasible set is convex, the Karush-Kuhn-Tucer conditions associated to the optimization problem (2-(4 are as follows: ( I c T vn p i=1 q vnit vni x ni ρ vnit α x ni τ t γnit 0; (40 q vnit ( I c T vn p i=1 q vnit vni x ni ρ vnit α x ni τ t γnit qvnit = 0; qvnit 0 q vnit [ z nit V v=1 q vnit ] 0; [ z nit V v=1 q vnit ] γ nit = 0; γ nit 0 (41 for v = 1,..., V, n = 1,..., N, i = 1,..., I, t = 1,..., T. Complementarity condition (40 has the following interpretation: if for v = 1,..., V, n = 1,..., N, i = 1,..., I, t = 1,..., T, it holds that q vnit > 0 then ( I c T vn p i=1 q vnit vni = x ni ρ vnit α x ni τ t γnit (42 q vnit A similar method can be applied to obtain prices p rmi, peco, p neco, and p m by using variational inequality (38 (or (19, (30, and (37. As indicated by [14], a variational inequality admits at least one solution if the feasible region Ω is compact and convex and the function F in (39 is continuous on Ω. In our model, we can guarantee the continuity of function F in (39, but not the compactness of the feasible region Ω. For this reason, we cannot derive existence of a solution simply from the assumption of continuity of the functions. However, as illustrated in [16], it is possible to introduce weaer conditions to guarantee solution existence. Let: Ω b = {(Q R, Q V, Q eco, Q neco, Q, p eco, p neco, γ, µ eco, µ neco, η eco, η neco, δ, σ, ϕ, λ 0 Q R b R ; 21

22 0 Q V b V ; 0 Q eco b eco ; 0 Q neco b neco ; 0 Q b m ; 0 p eco b eco p ; 0 p neco b neco p ; 0 γ b γ; 0 µ eco b eco µ ; 0 µ neco b neco µ ; 0 η eco b eco η ; 0 η neco b neco η ; 0 δ b δ ; 0 σ b σ; 0 ϕ b ϕ; 0 λ b λ }, where and b = (b R, b V, b eco, b neco, b m, b eco p, b neco p, b γ, b eco µ, b neco µ, b eco η, b neco η, b δ, b σ, b ϕ, b λ 0, 0 Q R b R ; 0 Q V b V ; 0 Q eco b eco ; 0 Q neco b neco ; 0 Q b m ; 0 p eco b eco p ; 0 p neco b neco p ; 0 γ b γ; 0 µ eco b eco µ ; 0 µ neco b neco µ ; 0 η eco b eco η ; 0 η neco b neco 0 δ b δ ; 0 σ b σ; 0 ϕ b ϕ; 0 λ b λ means that q rmit b R ; q vnit b V ; qt eco b eco ; qt neco b neco ; q m b m ; p eco b eco µ ; µ neco i b neco µ ; ηi eco b eco η ; ηi neco γ nit b γ; µ eco i λ mit b λ v, r, n, i,, m, t. b eco p ; p neco b neco p ; b neco η ; δ t b 8; σ b σ; ϕ rm b ϕ; Under these assumptions, Ω b is a bounded, closed, convex subset of RMIT V NIT IKT IKT KMKKNIT IIIIIKT KRMMIT R and the following variational inequality (43 F (X b, X X b 0, X b Ω b (43 admits at least one solution X b Ω b from the standard theory of variational inequalities, since Ω b is compact and F is continuous. Moreover, following [11], we then have: Lemma 1 Variational inequality (38 admits a solution if and only if there exists a b > 0 such that variational inequality (43 admits a solution in Ω b with Q R < b R ; Q V < b V ; Q eco < b eco ; Q neco < b neco ; Q < b m ; p eco < b eco p ; p neco < b neco p ; η ; γ < b γ; µ eco < b eco µ ; µ neco < b neco µ ; η eco < b eco η ; η neco < b neco η ; δ < b δ ; σ < b σ; ϕ < b ϕ; λ < b λ. Theorem 2 (Existence. Suppose that there exist positive constants A, B, C with C > A, such that: 22

23 c ri ( Mm=1 T q rmit q rmit ( Ii=1 c T vn q vnit c eco q vnit ( T qt eco q eco t ( c neco T qt neco q t neco tc eco x mi ρ rmit α x mi τ t ϕ rm λ mit η eco i A, Q R with q rmit B r, m, i, t (44 x ni ρ vnit α x ni τ t γ nit η eco i A, Q V with q vnit B r, n, i, t tc neco ( T qt eco q t eco ( T qt neco q t neco p CO2 φ eco i x ρ t α x τ t µ eco i η eco i δ t p eco θ eco σ M, Q eco with q eco t B p CO2 φ neco i x ρ t α x τ t µ neco i η neco i i,, t δ t p neco θ neco σ M, Q neco with q neco t B i,, t R a (Q θ neco c rm(β rm K σ s m =1 q m ω m β m x m ρ m x m ρ m β m β rm ϕ rm A, r=1 q m Q with q m B, m tc eco q eco t A, Q eco with q eco t B i,, t tc neco q neco t A, Q neco with q neco t B i,, t a (Q A, Q with q m B, m d eco (peco B, p eco with p eco > C d neco (p neco B, p neco with p neco > C Then variational inequality (38, or equivalently variational inequality (39, admits at least one solution. Under the condition of Theorem 2, it is possible to construct b R, b V, b eco, b neco, b m, b eco p, b neco p, b γ, b eco µ, b neco µ, b eco η, b neco η, b δ, b σ, b ϕ, b λ large enough so that variational inequality (43 will satisfy the boundedness condition described in Lemma 1. Uniqueness is another important property that one has to assure in order to guarantee the converge of an algorithm applied to solve any variational inequality. Theorem 3 (Strict monotonicity The vector function F of variational inequality (38 is strictly monotone, with respect to X = (Q R, Q V, Q eco, Q neco, Q, p eco, p neco, that is F (X F (X, X X > 0, X, X Ω, X X (45 under these two conditions: 1. One of the families of convex function c vn, c eco, cneco, c ri, c rm v, n, r, i,, m is a family of strictly convex functions; ( T ( T 2. tc eco qeco t ( i, ; tc neco qneco t ( i, ; a (Q (, m; d eco ( and d neco ( are strictly monotone. Theorem 4 (Uniqueness Under the conditions of Theorem 3, there must exist a unique reusable raw material supply pattern (Q R, a unique virgin raw material supply pattern (Q V, unique production patterns (Q eco and Q neco, a unique retail shipment (consumption pattern (Q, and unique demand price vectors (p eco and p neco satisfying 23